SEAMO X 2022 — Paper D — Grade 7–8

Let $A=\sum_{k=1}^{2022}\frac1{k\sqrt{k+1}+(k+1)\sqrt k}$. Suppose $A=a+b/\sqrt c$, with integers a,b,c and c divisible by no perfect square greater than one. Find a+b+c.

Distinct letters denote distinct digits in $MATH+MATH=HABIT$. Find HABIT.

Ten lines are concurrent. An eleventh line is parallel to one of them. How many distinct triangles do these lines form?

For nonzero real a,b,c, how many values can $a/|a|+b/|b|+c/|c|$ take?

Find the least positive integer k for which $2^{83}+k$ is divisible by one hundred twenty-seven.

For an arithmetic progression, let $S_n$ be its first n terms’ sum. Given $S_{15}\gt 0,S_{16}\lt 0$, define $x_i=S_i/a_i$ for $1\le i\le15$. For which i is x_i greatest?

Find the sum of the even positive divisors of ten thousand.

Rhombus ABCD has triangles ABC and ABD with circumradii thirty cm and ten cm respectively. Find its area.

Find the sum of all four-digit integers formed without repetition from digits zero, one, two, three and four.

Let d(n) be the units digit of $1+2+\cdots+n$. Find $d(1)+d(2)+\cdots+d(2022)$.

Fold an eight-by-fifteen rectangle so that two diagonally opposite vertices coincide. Find the crease length.

Primes p,q,r satisfy $pqr=19(p+q+r)$. Find $p^2+q^2+r^2$.

Arrange all distinct permutations of LETTER alphabetically. Find the one hundredth word.

In triangle ABC, AB=AC and angle BAC is one hundred twenty degrees. D is midpoint of BC; E is on AB with DE perpendicular to AB. Find AE:BD.

Integers $a_1,\ldots,a_n$, with $n\gt 1$, have both sum and product one hundred eleven. Find the least n.